In this article we study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic space $\B_n$. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution. In doing so, we put forward different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $\B_n$. We then study Hardy spaces $H^p(\B_n)$, $0< p <\infty$, whose elements appear as the hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere $H^p(\S^{n-1})$. In particular, we obtain an atomic decomposition of this spaces.
Publié le : 1999-07-05
Classification:
real hyperbolic ball,
harmonic functions,
boundary values,
Hardy spaces,
atomic decomposition,
48A85, 58G35,
[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]
@article{hal-00005820,
author = {Jaming, Philippe},
title = {Harmonic Functions on the Real Hyperbolic Ball I : Boundary Values and Atomic Decompositions of Hardy Spaces},
journal = {HAL},
volume = {1999},
number = {0},
year = {1999},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00005820}
}
Jaming, Philippe. Harmonic Functions on the Real Hyperbolic Ball I : Boundary Values and Atomic Decompositions of Hardy Spaces. HAL, Tome 1999 (1999) no. 0, . http://gdmltest.u-ga.fr/item/hal-00005820/